# The sum ∑ i = 1 n [ ∑ j = 1 n ( i + j ) ] .

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter F, Problem 50E
To determine

## To evaluate: The sum ∑i=1n[∑j=1n(i+j)].

Expert Solution

The value of the sum i=1n[j=1n(i+j)] is nm2(m+n+2).

### Explanation of Solution

Theorem used:

Let c be a constant and n be a positive integer. Then,

i=1nc=nc, i=1ni=n(n+1)2 and i=1ni2=n(n+1)(2n+1)6.

Calculation:

Simplify the expression i=1n[j=1n(i+j)] and obtain the value of the sum.

i=1m[j=1n(i+j)]=i=1n[j=1ni+j=1nj]=i=1n[ni+n(n+1)2][i=1nc=ncand i=1ni=n(n+1)2]=i=1nni+i=1nn(n+1)2=nm(m+1)2+nm(n+1)2=nm2(m+n+2)

Thus, the value of the sum i=1n[j=1n(i+j)] is nm2(m+n+2).

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